Definition

For a category , a morphism is called an isomorphism if there exists a morphism such that

Properties of isomorphisms

If is a morphism, then given the properties above is unique:

Let be an isomorphism and let be inverse functions such that

This means that . Using (associative) composition again with gives

Therefore, the inverse function is unique, and we can denote it by .

If is an isomorphism, then so too is and .

Let be an isomorphism. Stated again, this means that such that

g \circ f^{-1} = \text{id}_Y \quad \text{and} \quad f^{-1} \circ g = \text{id}_X$$ Let , thus .

If and are isomorphisms, then so too is .

Let and be isomorphisms. Then there exist morphisms and such that

Then for the morphism , consider the morphism .

and

Therefore and is a morphism.